Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type

• 成果类型: Article
• 署名作者: Kotelenez, Peter M.; Kurtz, Thomas G.
• 署名单位: University System of Ohio; Case Western Reserve University; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0188-0
• 发表日期: 2010
• 页码: 189-222
• 关键词: triangular arrays

摘要

A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. We also compare our approach to SODEs with the one introduced by Kunita.